I failed a course, and the system became my study supervisor.

Chapter 97 Solving Cascade Equations

Of course, it is impossible to participate in this project just for fun, and Xu Ling himself would not allow it.

Soon, Xu Ling and Yi Lian, as newcomers, received their first task together, which was to solve the cascade equation.

The reason why this task was assigned to them was largely because Senior Brother Dolson also disdained this issue.

I am still busy studying non-standard models with Professor Smith.

But as a newcomer, Xu Ling is aware of the situation and has no complaints.

He believed that with his abilities, it was only a matter of time before he could contribute to the core issues of the laboratory. His current task was to do the task at hand diligently and demonstrate his own value.

The evolution of particle cascades in the atmosphere can be described by the coupled cascade equations:
where Φi(E, X)dE is the flux of particles of type i with energies in the interval E to E + dE at the atmospheric slant depth X.

The first two terms on the right side of the equation are loss terms due to the interaction and decay of particle i, which are controlled by the interaction lengths λi int and λi dec.

The last two terms are source terms due to the interaction and decay of j-type particles, where dn/dE is the energy spectrum of the included particles.

For an observation height h in the atmosphere, the slant depth X along the locus s of the cascade center is given by:

……

There are eight equations in total. By solving them layer by layer, we can finally get the muon rate with energy Eμ.

But this is not the end. We will continue to study the relationship between the muon rate and the energy spectrum produced, with the focus on obtaining the muon production energy spectrum.

There are roughly three methods for this. The first method is to find an approximate solution to the cascade equation including pion and kaon channels.

The second is to find a numerical solution to the cascade equations involving all relevant channels.

A third, conceptually distinct approach is based on parameterizations of the muon production distribution in extended atmospheric showers, which are integrated over the primary particle flux.

"Xu, you solve the approximate and numerical solutions, and I will do the parameterization."

Xu Ling was about to show off his skills when Elaine made a suggestion.

Just pick the hardest one?

Xu Ling asked in surprise:

"why?"

“I think I might be more efficient.”

Elaine said bluntly.

As a student from the Sorbonne University, Elaine is very confident in her mathematical ability and does not think that someone of the same age can surpass her in this area.

But her arrogant words made Xu Ling a little unhappy:
"I do not think so!"

Are you kidding me about Math lv.2?

"Then why don't we have a competition to see who can use the parameterized method to obtain the muon production energy spectrum faster!"

Elaine said very casually.

Now, Xu Ling's competitive spirit was completely aroused.

No matter who I lose to, I can never lose to this French woman floating in the sky in front of me!

Xu Ling waited with a serious expression.

At the same time, the atmosphere between the two became somewhat tense and full of gunpowder.

The differential muon produces an energy spectrum P(Eμ,θ,X)=∫Eμ g(Eμ,E0,θ,X,T)ΦN(E0)dE0.

g(Eμ,E0,θ,X,T)=d(dN(……)/dX)/dEμ.

Here, dN(…)/dX is the average number of muons of energy Eμ produced per gram per square centimeter in a cosmic ray air shower initiated by a primary nucleus of mass number A, primary energy E0, and zenith angle θ.

It is a function of the slant depth X, where the atmospheric temperature at X is T.

……

Xu Ling's brain was working rapidly, and the pen in his hand did not stop at all. Equations and corresponding control conditions instantly filled the entire sheet of paper.

But the calculation was not over yet, and Yu Guangzhong and Elaine were also making rapid progress, so Xu Ling did not dare to relax at all.

A series of operations such as differentiation, integration, Laplace transform, etc. are smooth.

Just a few minutes later, when the last mathematical symbol was written, Xu Ling got the final result:

在给定天顶角的条件下，每次簇射的平均μ子数的近似值为N(Eμ)≈AK/Eμ cos(θ)(E0/AEμ)^α1(1-AEμ/E0)^α2。

At the same time, Elaine beside her raised her head. She had also solved it.

I rely on!

Your math is also level 2?

Xu Ling was a little shocked for a moment.

He is about the same age as me, and without any systematic support, his math ability is surprisingly similar to mine.

This really is the capital for being unrestrained by one's talent!

But no matter what, Elaine hit a wall today.

She thought she could leave Xu Ling far behind, but unexpectedly, they both finished at the same time.

"I'm sorry, I underestimated you!"

Elaine was filled with frustration.

For her, failure to win is failure.

"Haha, it's okay, you are really amazing."

Xu Ling said calmly.

However, this light tone seemed slightly mocking to Elaine.

In an instant, Elaine's face turned red, and she looked at Xu Ling with some resentment.

As a civilized gentleman, Xu Ling noticed that she was in a bad mood and immediately comforted her and explained that he had no other intentions.

At the same time, Xu Ling also took on the task of finding approximate solutions and numerical solutions.

Solving approximate solutions and solving using parameters are quite similar, and more lies in formula derivation.

But the numerical solution is different, because the formula is too complicated, and manual calculation is a thankless task. In the end, it still has to be calculated by computer through modeling.

After solving the cascade equations, the next step is to determine the effective temperature and correlation coefficient. Both of these have a characteristic: they are simple in concept but very complex in practice.

It took Xu Ling and Elaine two or three days to finish all the content.

"Are all Chinese students good at math?"

Elaine started chatting.

In the eyes of Westerners, every Chinese student is proficient in mathematics, especially numerical calculations.

There is an exaggerated saying that if a Chinese student and a calculator come up with two different answers, they will choose to believe the Chinese student without hesitation.

Elaine didn't believe it at first, but after seeing Xu Ling's mathematical ability, she became a little shaken.

Huh? Are Chinese students necessarily good at math?

"Who said that? That's just a stereotype."

Xu Ling was speechless.

While there are positive aspects to this stereotype, what about the others?
For example, appearance, height?
"People also say that the French are very arrogant and look down on those who can't speak French. Is that true?"

Xu Ling asked back.

"How could that be? Most of us are warm and open-minded, okay? How could we look down on others?"

Elaine hurried to explain.

"See, this is also a stereotype," Xu Ling said in a didactic tone. "Don't label people based on their ethnic stereotypes."

"Why do you speak so old-fashionedly?"

Elaine curled her lips.

"Have it?"

"Yes, Grandpa Xu!"

Ok?
Why call me grandpa?

This foreigner is so open-minded!

……

(End of this chapter)